# nLab Sandbox

Idea

## Idea

Expositions of topological data analysis traditionally invoke point cloudsdiscrete subsets of some metric space – as the generic mathematical incarnation of datasets to be analyzed.

Maybe a more realistic and more encompassing model for sets of observed data – namely for measurement results – is the time-honored notion of a tuple of (values of) real observables, namely a continuous function

$Obs \;\colon\; X \xrightarrow{\phantom{---}} \mathbb{R}^n$

to the one-point compactification of $n$-dimensional Cartesian space.

A feature in the data as seen by such an observable is then an isosurface of $Obs$, hence the pre-image

$Obs^{-1}(\vec v) \;\coloneqq\; \Big\{ x \in X \,\big\vert\, Obs(x) = \vec v \Big\} \;\subset\; X$

of a tuple $\vec v \in \mathbb{R}^n$ of observed values.

Without restriction of generality we may assume that the observed value of interest is the origin $0 \in \mathbb{R}^n$, for if it is instead some $\vec v \in \mathbb{R}^n$ then we may instead pass to the observable $Obs \coloneqq Obs - const_{\vec v}$ without changing the essence of the situation.

In practice, the value of an observable can never be determined with the accuracy of a mathematical point in $\mathbb{R}^n$, instead there will be some positive real number $r \in \mathbb{R}_{\gt 0}$ such that one may hope (or wish) to measure $Obs$ up to measurement errors within a radius $r$. In this case, the desired isosurface could be any element in the set

$\Big\{ f^{-1}(0) \subset X \,\vert\, f \in C^0(X,\mathbb{R}^n) ,\, \Vert f - Obs \Vert_\infty \lt r \Big\}$

For example, $X$ might model the interior of a plasma container (say a fusion reactor) and $Ob = (T, p) : X \to \mathbb{R}^2$ could be the combined temperature- and pressure-observable (say as seen by laser probes into the plasma). Its isosurfaces are the intersections of given isobars with isotherms?.

Illustrating the main theorem of persistent Cohomotopy.

Suppose we want to know the zeros of the observable $f$. If the resolution/error bar of measuring $f$ is $r$, then we know that

1. the zeros must be somewhere away from the gray region.

2. the homotopy class of $f$ on the quotient determines the number of zeros mod 2.

Last revised on May 21, 2022 at 16:05:43. See the history of this page for a list of all contributions to it.